Family Of Polytopes Research Articles

Neighborly inscribed polytopes and delaunay triangulations

Abstract We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that for d ≥ 4 there are superexponentially many combinatorially distinct neighborly d-polytopes on n vertices that admit realizations inscribed in the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations in ℝ d for d ≥ 3.

Common Information and Unique Disjointness

We provide an information-theoretic framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in Wyner (IEEE Trans Inf Theory 21(2):163---179, 1975). The framework is a generalization of the one in Braverman and Moitra (Proceedings of the forty-fifth annual ACM symposium on theory of computing, pp 161---170, 2013) for the shifted uniqe disjointness (UDISJ) matrix to arbitrary nonnegative matrices. Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with Hellinger distance estimations we compute the (almost) exact common information of UDISJ partial matrix. The bounds are obtained very naturally. We also establish robustness of this estimation under random or adversarial removal of rows and columns of the UDISJ partial matrix. This robustness translates, via a variant of Yannakakis's factorization theorem, to lower bounds on the average case and adversarial approximate extension complexity of removals. We present the first family of polytopes, the hard pair introduced in Braun et al. (Math Oper Res 40(3):756---772, 2015) related to the CLIQUE problem, with high average case and adversarial approximate extension complexity of removals. The framework relies on a strengthened version of the link between information theory and Hellinger distance from Bar-Yossef (J Comput Syst Sci 68(4):702---732, 2004). We also provide an information theoretic variant of the fooling set method that allows us to extend fooling set lower bounds from extension complexity to approximate extension complexity.

A Special Role of Boolean Quadratic Polytopes among Other Combinatorial Polytopes

We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we use the following method. We say that a family

On the diameter of cut polytopes

Given an undirected graph G with node set V and edge set E, the cut polytope CUT(G) is defined as the convex hull of the incidence vectors of the cuts in G. For a given integer k∈{1,2,…,⌊|V|2⌋}, the uniform cut polytope CUT=k(G) is defined as the convex hull of the incidence vectors of the cuts which correspond to a bipartition of the node set into sets with cardinalities k and |V|−k.In this paper, we study the diameter of these two families of polytopes. For a connected graph G, we prove that the diameter of CUT(G) is at most |V|−1, improving on the bound of |E| given by F. Barahona and A.R. Mahjoub. We also give its exact value for trees and complete bipartite graphs. Then, with respect to uniform cut polytopes, we introduce sufficient conditions for adjacency of two vertices in CUT=k(G), we study some particular cases and provide some connections with other partition polytopes from the literature.

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay($\hat{G}$,S)) with maximal cliques related to T. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G. We apply our general result to two examples. First, in the case where $G = \mathbb{Z}_2^n$, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most $\lceil n/2 \rceil$, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on $\mathbb{Z}_N$ (when d divides N). By constructing a particular chordal cover of the d'th power of the N-cycle, we prove that any such function is a sum of squares of functions with at most $3d\log(N/d)$ nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in $\mathbb{R}^{2d}$ with N vertices can be expressed as a projection of a section of the cone of psd matrices of size $3d\log(N/d)$. Putting $N=d^2$ gives a family of polytopes $P_d \subset \mathbb{R}^{2d}$ with LP extension complexity $\text{xc}_{LP}(P_d) = \Omega(d^2)$ and SDP extension complexity $\text{xc}_{PSD}(P_d) = O(d\log(d))$. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where $\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))$.

Pipe Dream Complexes and Triangulations of Root Polytopes Belong Together

We show that the pipe dream complex associated to the permutation $1\text{ } n \text{ }n-1\text{ } \cdots \text{ }2$ can be geometrically realized as a triangulation of the vertex figure of a root polytope. Leading up to this result we show that the Grothendieck polynomial specializes to the $h$-polynomial of the corresponding pipe dream complex, which in certain cases equals the $h$-polynomial of canonical triangulations of root (and flow) polytopes, which in turn equals a specialization of the reduced form of a monomial in the subdivision algebra of root (and flow) polytopes. Thus, we connect Grothendieck polynomials to reduced forms in subdivision algebras and root (and flow) polytopes. We also show that root polytopes can be seen as projections of flow polytopes, explaining that these families of polytopes possess the same subdivision algebra.

INTERMEDIATE SUMS ON POLYHEDRA II: BIDEGREE AND POISSON FORMULA

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s + c parallel to the subspace L and sum up the integrals. We study these intermediate sums by means of the intermediate generating functions $S^L(s+c)(\xi)$, and expose the bidegree structure in parameters s and $\xi$, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra, Found. Comput. Math. 12 (2012), 435-469] and [Intermediate sums on polyhedra: Computation and real Ehrhart theory, Mathematika 59 (2013), 1-22]. The bidegree structure is key to a new proof for the Baldoni--Berline--Vergne approximation theorem for discrete generating functions [Local Euler--Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes, Contemp. Math. 452 (2008), 15-33], using the Fourier analysis with respect to the parameter s and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

Reliability of Erasure Coded Storage Systems: A Combinatorial-Geometric Approach

We consider the probability of data loss, or equivalently, the reliability function for an erasure coded distributed data storage system under worst case conditions. Data loss in an erasure coded system depends on probability distributions for the disk repair duration and the disk failure duration. In previous works, the data loss probability of such systems has been studied under the assumption of exponentially distributed disk failure and disk repair durations, using well-known analytic methods from the theory of Markov processes. These methods lead to an estimate of the integral of the reliability function. Here, we address the problem of directly calculating the data loss probability for general repair and failure duration distributions. A closed limiting form is developed for the probability of data loss and it is shown that the probability of the event that a repair duration exceeds a failure duration is sufficient for characterizing the data loss probability. For the case of constant repair duration, we develop an expression for the conditional data loss probability given the number of failures experienced by a each node in a given time window. We do so by developing a geometric approach that relies on the computation of volumes of a family of polytopes that are related to the code. An exact calculation is provided and an upper bound on the data loss probability is obtained by posing the problem as a set avoidance problem. Theoretical calculations are compared to simulation results.

On the metric dimension of two families of convex polytopes

A distance between two vertices of a connected graph is the shortest distance between them. The metric dimension of a connected graph G is the minimum cardinality of a subset W of vertices of G such that all vertices are uniquely determined by their distances from W. A family $$\mathcal {G}$$ of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in $$\mathcal {G}$$ . In this paper we study the metric dimension of two classes of convex polytopes and show that these classes of convex polytopes have constant metric dimension.

Ehrhart Positivity for Generalized Permutohedra

There are few general results about the coefficients of Ehrhart polynomials. We present a conjecture about their positivity for a certain family of polytopes known as generalized permutohedra. We have verified the conjecture for small dimensions combining perturbation methods with a new valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne. Il existe peu de résultats sur les coefficients des polynômes d’Ehrhart. On présente une conjecture concernant leur positivité pour une certaine famille de polytopes connus sous le nom de permutoèdre généralisé. On a vérifié la conjecture pour les petites dimensions en combinant des méthodes de perturbation avec une nouvelle valuation sur l’algèbre des cônes polyédraux rationnels pointés, construite par Berline et Vergne.

The simplest families of polytopes associated with NP-hard problems

The simplest families of polytopes associated with NP-hard problems

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

A central question in optimization is to maximize (or minimize) a linear function over a given polytope P. To solve such a problem in practice one needs a concise description of the polytope P. In this paper we are interested in representations of P using the positive semidefinite cone: a positive semidefinite lift (psd lift) of a polytope P is a representation of P as the projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. Such a representation allows linear optimization problems over P to be written as semidefinite programs of size d. Such representations can be beneficial in practice when d is much smaller than the number of facets of the polytope P. In this paper we are concerned with so-called equivariant psd lifts (also known as symmetric psd lifts) which respect the symmetries of the polytope P. We present a representation-theoretic framework to study equivariant psd lifts of a certain class of symmetric polytopes known as orbitopes. Our main result is a structure theorem where we show that any equivariant psd lift of size d of an orbitope is of sum-of-squares type where the functions in the sum-of-squares decomposition come from an invariant subspace of dimension smaller than d^3. We use this framework to study two well-known families of polytopes, namely the parity polytope and the cut polytope, and we prove exponential lower bounds for equivariant psd lifts of these polytopes.

Product formulas for volumes of flow polytopes

Intrigued by the product formula prod_{i=1}^{n-2} C_i for the volume of the Chan-Robbins-Yuen polytope CRY_n, where C_i is the ith Catalan number, we construct a family of polytopes P_{m,n}, whose volumes are given by the product \prod_{i=m+1}^{m+n-2}\frac{1}{2i+1}{{m+n+i} \choose {2i}}. The Chan-Robbins-Yuen polytope CRY_n coincides with P_{0,n-1}. Our construction of the polytopes P_{m,n} is an application of a systematic method we develop for expressing volumes of a class of flow polytopes as the number of certain triangular arrays. This method can also be used as a heuristic technique for constructing polytopes with combinatorial volumes. As an illustration of this we construct polytopes whose volumes equal the number of r-ary trees on n internal nodes, \frac{1}{(r-1)n+1} {{rn} \choose n}. Using triangular arrays we also express the volumes of flow polytopes as constant terms of formal Laurent series.

All Clauser–Horne–Shimony–Holt polytopes

The correlations that admit a local hidden-variable model are described by a family of polytopes, whose facets are the Bell inequalities. The Clauser–Horne–Shimony–Holt (CHSH) inequality is the simplest such Bell inequality and is a facet of every Bell polytope. We investigate for which Bell polytopes the CHSH inequality is also the unique (non-trivial) facet. We prove that the CHSH inequality is the unique facet for all bipartite polytopes where at least one party has a binary choice of dichotomic measurements, irrespective of the number of measurement settings and outcomes for the other party. Based on numerical results, we conjecture that it is also the unique facet for all bipartite polytopes involving two measurements per party where at least one measurement is dichotomic. Finally, we remark that these two situations can be the only ones for which the CHSH inequality is the unique facet, i.e., any polytope that does not correspond to one of these two cases necessarily has facets that are not of the CHSH form. As a byproduct of our approach, we derive a new family of facet inequalities.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.

Extremal Edge Polytopes

The "edge polytope" of a finite graph $G$ is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For $k =2, 3, 5$ we determine the maximal number of vertices of $k$-neighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.

A Helly-type theorem for intersections of orthogonally starshaped sets in $$\mathbb R^d$$ R d

Let \(\mathcal K\) be a finite family of orthogonal polytopes in \(\mathbb R^d\) such that, for every nonempty subfamily \(\mathcal K^\prime \) of \(\mathcal K, \cap \{K : K\) in \(\mathcal K^\prime \}\), if nonempty, is a finite union of boxes whose intersection graph is a tree. Assume that every \(d + 1\) (not necessarily distinct) members of \(\mathcal K\) meet in a (nonempty) staircase starshaped set. Then \(S \equiv \cap \{ K : K\) in \(\mathcal K\}\) is nonempty and staircase starshaped.

On Counterexamples to a Conjecture of Wills and Ehrhart Polynomials whose Roots have Equal Real Parts

As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a $0$-symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side length two. We discuss several counterexamples to this conjecture and, on the positive side, we identify a family of lattice polytopes that fulfill the claimed inequalities. This family is related to the recently introduced class of $l$-reflexive polytopes.

An approach of the minimal model program for horospherical varieties via moment polytopes

Abstract We describe the minimal model program in the family of ℚ-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP for toric varieties due to M. Reid, and we complete the results on MMP for spherical varieties due to M. Brion in the case of horospherical varieties.

Flow Polytopes of Signed Graphs and the Kostant Partition Function

We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an interesting special family of flow polytopes, we study the Chan–Robbins–Yuen (CRY) polytopes. Motivated by the volume formula for the type An version, where Cat(k) is the kth Catalan number, we introduce type Cn+1 and Dn+1 CRY polytopes along with intriguing conjectures about their volumes.

On stretching the interval simplex-permutohedron

A family of polytopes introduced by E.M. Feichtner, A. Postnikov, and B. Sturmfels, which were named nestohedra, consists in each dimension of an interval of polytopes starting with a simplex and ending with a permutohedron. This paper investigates a problem of changing and extending the boundaries of these intervals. An iterative application of Feichtner---Kozlov procedure of forming complexes of nested sets is a solution of this problem. By using a simple algebraic presentation of members of nested sets it is possible to avoid the problem of increasing the complexity of the structure of nested curly braces in elements of the produced simplicial complexes.